A Strongly Polynomial Algorithm for a Class of Minimum-Cost Flow Problems with Separable Convex Objectives

Abstract

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E}C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$'s are convex quadratic functions, settling an open problem raised, e.g., by Hochbaum [Math. Oper. Res., 19 (1994), pp. 390--409]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities that can be formulated in this framework (see Shmyrev [J. Appl. Ind. Math., 3 (2009), pp. 505--518], Birnbaum, Devanur, and Xiao [Proceedings of the 12th ACM Conference on Electronic Commerce, 2011, pp. 127--136]). For the latter class this resolves an open question raised by Vazirani [Math. Oper. Res., 35 (2010), pp. 458--478]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher's markets with linear utilities, and $O(mn^3+m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [Math. Oper. Res., 19 (1994), pp. 390--409]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings.